We view \({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  as endowed with the Kan-Quillen model structure.
The category \({{C}\mathrm {art}}\)  carries a Grothendieck pretopology \(\tau \)  of differentiably good open coverings—see [1]}, [2]} for details.
A covering of \(c \in {{C}\mathrm {art}}\)  in this pretopology is a collection of morphisms \(\lbrace \iota _i \colon c_i \rightarrow c\rbrace _{i \in \Lambda }\)  in \({{C}\mathrm {art}}\)  such that each \(\iota _i\)  is an embedding of an open subset, the images of the maps \(\iota _i\)  cover \(c\)  (i.e. each \(x \in c\)  lies in the image of some \(\iota _i\) ), and every finite intersection
\(C_{i_0 \ldots i_n} \bigcap _{a = 0}^n \iota _{i_a}(c_{i_a}) \subset c\) 
